X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=5dee5d012d6c19d6fc5d79f804ad8aff2757c3d0;hb=ea99a21239882d478c1a458a0411b1eb0588b84b;hp=ec3dd1141a9f497acca26f6d09c830420dfb5d5d;hpb=4e966338c00181067c4d10f24b3fcaf288ebe207;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index ec3dd11..5dee5d0 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -119,10 +119,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): The ``field`` we're given must be real with ``check_field=True``:: - sage: JordanSpinEJA(2,QQbar) + sage: JordanSpinEJA(2, field=QQbar) Traceback (most recent call last): ... ValueError: scalar field is not real + sage: JordanSpinEJA(2, field=QQbar, check_field=False) + Euclidean Jordan algebra of dimension 2 over Algebraic Field The multiplication table must be square with ``check_axioms=True``:: @@ -291,6 +293,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: x = J.random_element() sage: J(x.to_vector().column()) == x True + """ msg = "not an element of this algebra" if elt == 0: @@ -313,8 +316,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # element's ring because the basis space might be an algebraic # closure whereas the base ring of the 3-by-3 identity matrix # could be QQ instead of QQbar. + # + # We pass check=False because the matrix basis is "guaranteed" + # to be linearly independent... right? Ha ha. V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols()) - W = V.span_of_basis( _mat2vec(s) for s in self.matrix_basis() ) + W = V.span_of_basis( (_mat2vec(s) for s in self.matrix_basis()), + check=False) try: coords = W.coordinate_vector(_mat2vec(elt)) @@ -393,7 +400,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # Used to check whether or not something is zero in an inexact # ring. This number is sufficient to allow the construction of - # QuaternionHermitianEJA(2, RDF) with check_axioms=True. + # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True. epsilon = 1e-16 for i in range(self.dimension()): @@ -490,7 +497,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = HadamardEJA(2) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X1, X2... - sage: J = RealSymmetricEJA(3,QQ,orthonormalize=False) + sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False) sage: J.coordinate_polynomial_ring() Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6... @@ -1155,16 +1162,26 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr """ def __init__(self, - field, basis, jordan_product, inner_product, + field=AA, orthonormalize=True, prefix='e', category=None, check_field=True, check_axioms=True): + if check_field: + # Abuse the check_field parameter to check that the entries of + # out basis (in ambient coordinates) are in the field QQ. + if not all( all(b_i in QQ for b_i in b.list()) for b in basis ): + raise TypeError("basis not rational") + + # Temporary(?) hack to ensure that the matrix and vector bases + # are over the same ring. + basis = tuple( b.change_ring(field) for b in basis ) + n = len(basis) vector_basis = basis @@ -1203,10 +1220,10 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr # because they are necessarily defined with respect to # ambient coordinates and not any particular basis. self._rational_algebra = RationalBasisEuclideanJordanAlgebra( - QQ, basis, jordan_product, inner_product, + field=QQ, orthonormalize=False, prefix=prefix, category=category, @@ -1214,8 +1231,9 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr check_axioms=False) # Compute the deorthonormalized tables before we orthonormalize - # the given basis. - W = V.span_of_basis( vector_basis ) + # the given basis. The "check" parameter here guarantees that + # the basis is linearly-independent. + W = V.span_of_basis( vector_basis, check=check_axioms) # Note: the Jordan and inner-products are defined in terms # of the ambient basis. It's important that their arguments @@ -1251,22 +1269,20 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr else: vector_basis = gram_schmidt(vector_basis, inner_product) - W = V.span_of_basis( vector_basis ) - # Normalize the "matrix" basis, too! basis = vector_basis if basis_is_matrices: basis = tuple( map(_vec2mat,basis) ) - W = V.span_of_basis( vector_basis ) + W = V.span_of_basis( vector_basis, check=check_axioms) # Now "W" is the vector space of our algebra coordinates. The # variables "X1", "X2",... refer to the entries of vectors in # W. Thus to convert back and forth between the orthonormal # coordinates and the given ones, we need to stick the original # basis in W. - U = V.span_of_basis( deortho_vector_basis ) + U = V.span_of_basis( deortho_vector_basis, check=check_axioms) self._deortho_matrix = matrix( U.coordinate_vector(q) for q in vector_basis ) @@ -1349,9 +1365,11 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr Algebraic Real Field """ - if self.base_ring() is QQ: + if self.base_ring() is QQ or self._rational_algebra is None: # There's no need to construct *another* algebra over the - # rationals if this one is already over the rationals. + # rationals if this one is already over the + # rationals. Likewise, if we never orthonormalized our + # basis, we might as well just use the given one. superclass = super(RationalBasisEuclideanJordanAlgebra, self) return superclass._charpoly_coefficients() @@ -1441,7 +1459,22 @@ class ConcreteEuclideanJordanAlgebra(RationalBasisEuclideanJordanAlgebra): class MatrixEuclideanJordanAlgebra: @staticmethod - def real_embed(M): + def dimension_over_reals(): + r""" + The dimension of this matrix's base ring over the reals. + + The reals are dimension one over themselves, obviously; that's + just `\mathbb{R}^{1}`. Likewise, the complex numbers `a + bi` + have dimension two. Finally, the quaternions have dimension + four over the reals. + + This is used to determine the size of the matrix returned from + :meth:`real_embed`, among other things. + """ + raise NotImplementedError + + @classmethod + def real_embed(cls,M): """ Embed the matrix ``M`` into a space of real matrices. @@ -1454,15 +1487,21 @@ class MatrixEuclideanJordanAlgebra: real_embed(M*N) = real_embed(M)*real_embed(N) """ - raise NotImplementedError + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + return M - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of :meth:`real_embed`. """ - raise NotImplementedError + if M.ncols() != M.nrows(): + raise ValueError("the matrix 'M' must be square") + if not ZZ(M.nrows()).mod(cls.dimension_over_reals()).is_zero(): + raise ValueError("the matrix 'M' must be a real embedding") + return M @staticmethod def jordan_product(X,Y): @@ -1470,36 +1509,65 @@ class MatrixEuclideanJordanAlgebra: @classmethod def trace_inner_product(cls,X,Y): + r""" + Compute the trace inner-product of two real-embeddings. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, + ....: ComplexHermitianEJA, + ....: QuaternionHermitianEJA) + + EXAMPLES:: + + This gives the same answer as it would if we computed the trace + from the unembedded (original) matrices:: + + sage: set_random_seed() + sage: J = ComplexHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = ComplexHermitianEJA.real_unembed(Xe) + sage: Y = ComplexHermitianEJA.real_unembed(Ye) + sage: expected = (X*Y).trace().real() + sage: actual = ComplexHermitianEJA.trace_inner_product(Xe,Ye) + sage: actual == expected + True + + :: + + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() + sage: X = QuaternionHermitianEJA.real_unembed(Xe) + sage: Y = QuaternionHermitianEJA.real_unembed(Ye) + sage: expected = (X*Y).trace().coefficient_tuple()[0] + sage: actual = QuaternionHermitianEJA.trace_inner_product(Xe,Ye) + sage: actual == expected + True + + """ Xu = cls.real_unembed(X) Yu = cls.real_unembed(Y) tr = (Xu*Yu).trace() try: # Works in QQ, AA, RDF, et cetera. - return tr.real() + return tr.real() / cls.dimension_over_reals() except AttributeError: # A quaternion doesn't have a real() method, but does # have coefficient_tuple() method that returns the # coefficients of 1, i, j, and k -- in that order. - return tr.coefficient_tuple()[0] + return tr.coefficient_tuple()[0] / cls.dimension_over_reals() class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @staticmethod - def real_embed(M): - """ - The identity function, for embedding real matrices into real - matrices. - """ - return M - - @staticmethod - def real_unembed(M): - """ - The identity function, for unembedding real matrices from real - matrices. - """ - return M + def dimension_over_reals(): + return 1 class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, @@ -1526,9 +1594,9 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: RealSymmetricEJA(2, RDF) + sage: RealSymmetricEJA(2, field=RDF) Euclidean Jordan algebra of dimension 3 over Real Double Field - sage: RealSymmetricEJA(2, RR) + sage: RealSymmetricEJA(2, field=RR) Euclidean Jordan algebra of dimension 3 over Real Field with 53 bits of precision @@ -1569,7 +1637,7 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Return a basis for the space of real symmetric n-by-n matrices. @@ -1581,7 +1649,7 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = RealSymmetricEJA._denormalized_basis(n,QQ) + sage: B = RealSymmetricEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B) True @@ -1591,7 +1659,7 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, S = [] for i in range(n): for j in range(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) + Eij = matrix(ZZ, n, lambda k,l: k==i and l==j) if i == j: Sij = Eij else: @@ -1605,27 +1673,39 @@ class RealSymmetricEJA(ConcreteEuclideanJordanAlgebra, return 4 # Dimension 10 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this type of algebra. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) - return cls(n, field, **kwargs) + return cls(n, **kwargs) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n, field) - super(RealSymmetricEJA, self).__init__(field, - basis, + def __init__(self, n, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n), self.jordan_product, self.trace_inner_product, **kwargs) + + # TODO: this could be factored out somehow, but is left here + # because the MatrixEuclideanJordanAlgebra is not presently + # a subclass of the FDEJA class that defines rank() and one(). self.rank.set_cache(n) - self.one.set_cache(self(matrix.identity(field,n))) + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) + class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @staticmethod - def real_embed(M): + def dimension_over_reals(): + return 2 + + @classmethod + def real_embed(cls,M): """ Embed the n-by-n complex matrix ``M`` into the space of real matrices of size 2n-by-2n via the map the sends each entry `z = a + @@ -1667,9 +1747,8 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): True """ + super(ComplexMatrixEuclideanJordanAlgebra,cls).real_embed(M) n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") # We don't need any adjoined elements... field = M.base_ring().base_ring() @@ -1683,8 +1762,8 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): return matrix.block(field, n, blocks) - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of _embed_complex_matrix(). @@ -1715,11 +1794,9 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): True """ + super(ComplexMatrixEuclideanJordanAlgebra,cls).real_unembed(M) n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(2).is_zero(): - raise ValueError("the matrix 'M' must be a complex embedding") + d = cls.dimension_over_reals() # If "M" was normalized, its base ring might have roots # adjoined and they can stick around after unembedding. @@ -1737,9 +1814,9 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): # Go top-left to bottom-right (reading order), converting every # 2-by-2 block we see to a single complex element. elements = [] - for k in range(n/2): - for j in range(n/2): - submat = M[2*k:2*k+2,2*j:2*j+2] + for k in range(n/d): + for j in range(n/d): + submat = M[d*k:d*k+d,d*j:d*j+d] if submat[0,0] != submat[1,1]: raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0]: @@ -1747,38 +1824,7 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): z = submat[0,0] + submat[0,1]*i elements.append(z) - return matrix(F, n/2, elements) - - - @classmethod - def trace_inner_product(cls,X,Y): - """ - Compute a matrix inner product in this algebra directly from - its real embedding. - - SETUP:: - - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA - - TESTS: - - This gives the same answer as the slow, default method implemented - in :class:`MatrixEuclideanJordanAlgebra`:: - - sage: set_random_seed() - sage: J = ComplexHermitianEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: Xe = x.to_matrix() - sage: Ye = y.to_matrix() - sage: X = ComplexHermitianEJA.real_unembed(Xe) - sage: Y = ComplexHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().real() - sage: actual = ComplexHermitianEJA.trace_inner_product(Xe,Ye) - sage: actual == expected - True - - """ - return RealMatrixEuclideanJordanAlgebra.trace_inner_product(X,Y)/2 + return matrix(F, n/d, elements) class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, @@ -1797,9 +1843,9 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: ComplexHermitianEJA(2, RDF) + sage: ComplexHermitianEJA(2, field=RDF) Euclidean Jordan algebra of dimension 4 over Real Double Field - sage: ComplexHermitianEJA(2, RR) + sage: ComplexHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 4 over Real Field with 53 bits of precision @@ -1841,7 +1887,7 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Returns a basis for the space of complex Hermitian n-by-n matrices. @@ -1860,15 +1906,16 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: field = QuadraticField(2, 'sqrt2') - sage: B = ComplexHermitianEJA._denormalized_basis(n, field) + sage: B = ComplexHermitianEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B) True """ + field = ZZ R = PolynomialRing(field, 'z') z = R.gen() F = field.extension(z**2 + 1, 'I') - I = F.gen() + I = F.gen(1) # This is like the symmetric case, but we need to be careful: # @@ -1894,31 +1941,41 @@ class ComplexHermitianEJA(ConcreteEuclideanJordanAlgebra, return tuple( s.change_ring(field) for s in S ) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n,field) - super(ComplexHermitianEJA, self).__init__(field, - basis, - self.jordan_product, - self.trace_inner_product, - **kwargs) + def __init__(self, n, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n), + self.jordan_product, + self.trace_inner_product, + **kwargs) + # TODO: this could be factored out somehow, but is left here + # because the MatrixEuclideanJordanAlgebra is not presently + # a subclass of the FDEJA class that defines rank() and one(). self.rank.set_cache(n) - # TODO: pre-cache the identity! + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) @staticmethod def _max_random_instance_size(): return 3 # Dimension 9 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this type of algebra. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) - return cls(n, field, **kwargs) + return cls(n, **kwargs) class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @staticmethod - def real_embed(M): + def dimension_over_reals(): + return 4 + + @classmethod + def real_embed(cls,M): """ Embed the n-by-n quaternion matrix ``M`` into the space of real matrices of size 4n-by-4n by first sending each quaternion entry `z @@ -1957,10 +2014,9 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): True """ + super(QuaternionMatrixEuclideanJordanAlgebra,cls).real_embed(M) quaternions = M.base_ring() n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") F = QuadraticField(-1, 'I') i = F.gen() @@ -1983,8 +2039,8 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): - @staticmethod - def real_unembed(M): + @classmethod + def real_unembed(cls,M): """ The inverse of _embed_quaternion_matrix(). @@ -2014,11 +2070,9 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): True """ + super(QuaternionMatrixEuclideanJordanAlgebra,cls).real_unembed(M) n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(4).is_zero(): - raise ValueError("the matrix 'M' must be a quaternion embedding") + d = cls.dimension_over_reals() # Use the base ring of the matrix to ensure that its entries can be # multiplied by elements of the quaternion algebra. @@ -2030,10 +2084,10 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 # quaternion block. elements = [] - for l in range(n/4): - for m in range(n/4): + for l in range(n/d): + for m in range(n/d): submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed( - M[4*l:4*l+4,4*m:4*m+4] ) + M[d*l:d*l+d,d*m:d*m+d] ) if submat[0,0] != submat[1,1].conjugate(): raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0].conjugate(): @@ -2044,38 +2098,7 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): z += submat[0,1].imag()*k elements.append(z) - return matrix(Q, n/4, elements) - - - @classmethod - def trace_inner_product(cls,X,Y): - """ - Compute a matrix inner product in this algebra directly from - its real embedding. - - SETUP:: - - sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA - - TESTS: - - This gives the same answer as the slow, default method implemented - in :class:`MatrixEuclideanJordanAlgebra`:: - - sage: set_random_seed() - sage: J = QuaternionHermitianEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: Xe = x.to_matrix() - sage: Ye = y.to_matrix() - sage: X = QuaternionHermitianEJA.real_unembed(Xe) - sage: Y = QuaternionHermitianEJA.real_unembed(Ye) - sage: expected = (X*Y).trace().coefficient_tuple()[0] - sage: actual = QuaternionHermitianEJA.trace_inner_product(Xe,Ye) - sage: actual == expected - True - - """ - return RealMatrixEuclideanJordanAlgebra.trace_inner_product(X,Y)/4 + return matrix(Q, n/d, elements) class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, @@ -2094,9 +2117,9 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, In theory, our "field" can be any subfield of the reals:: - sage: QuaternionHermitianEJA(2, RDF) + sage: QuaternionHermitianEJA(2, field=RDF) Euclidean Jordan algebra of dimension 6 over Real Double Field - sage: QuaternionHermitianEJA(2, RR) + sage: QuaternionHermitianEJA(2, field=RR) Euclidean Jordan algebra of dimension 6 over Real Field with 53 bits of precision @@ -2137,7 +2160,7 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, """ @classmethod - def _denormalized_basis(cls, n, field): + def _denormalized_basis(cls, n): """ Returns a basis for the space of quaternion Hermitian n-by-n matrices. @@ -2155,11 +2178,12 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ) + sage: B = QuaternionHermitianEJA._denormalized_basis(n) sage: all( M.is_symmetric() for M in B ) True """ + field = ZZ Q = QuaternionAlgebra(QQ,-1,-1) I,J,K = Q.gens() @@ -2192,15 +2216,22 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, return tuple( s.change_ring(field) for s in S ) - def __init__(self, n, field=AA, **kwargs): - basis = self._denormalized_basis(n,field) - super(QuaternionHermitianEJA, self).__init__(field, - basis, + def __init__(self, n, **kwargs): + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n), self.jordan_product, self.trace_inner_product, **kwargs) + # TODO: this could be factored out somehow, but is left here + # because the MatrixEuclideanJordanAlgebra is not presently + # a subclass of the FDEJA class that defines rank() and one(). self.rank.set_cache(n) - # TODO: cache one()! + idV = matrix.identity(ZZ, self.dimension_over_reals()*n) + self.one.set_cache(self(idV)) + @staticmethod def _max_random_instance_size(): @@ -2210,12 +2241,12 @@ class QuaternionHermitianEJA(ConcreteEuclideanJordanAlgebra, return 2 # Dimension 6 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this type of algebra. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) - return cls(n, field, **kwargs) + return cls(n, **kwargs) class HadamardEJA(ConcreteEuclideanJordanAlgebra): @@ -2258,17 +2289,25 @@ class HadamardEJA(ConcreteEuclideanJordanAlgebra): (r0, r1, r2) """ - def __init__(self, n, field=AA, **kwargs): - V = VectorSpace(field, n) - basis = V.basis() - + def __init__(self, n, **kwargs): def jordan_product(x,y): - return V([ xi*yi for (xi,yi) in zip(x,y) ]) + P = x.parent() + return P(tuple( xi*yi for (xi,yi) in zip(x,y) )) def inner_product(x,y): return x.inner_product(y) - super(HadamardEJA, self).__init__(field, - basis, + # New defaults for keyword arguments. Don't orthonormalize + # because our basis is already orthonormal with respect to our + # inner-product. Don't check the axioms, because we know this + # is a valid EJA... but do double-check if the user passes + # check_axioms=True. Note: we DON'T override the "check_field" + # default here, because the user can pass in a field! + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + + standard_basis = FreeModule(ZZ, n).basis() + super(HadamardEJA, self).__init__(standard_basis, jordan_product, inner_product, **kwargs) @@ -2287,12 +2326,12 @@ class HadamardEJA(ConcreteEuclideanJordanAlgebra): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this type of algebra. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) - return cls(n, field, **kwargs) + return cls(n, **kwargs) class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): @@ -2376,27 +2415,30 @@ class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): sage: actual == expected True """ - def __init__(self, B, field=AA, **kwargs): + def __init__(self, B, **kwargs): if not B.is_positive_definite(): raise ValueError("bilinear form is not positive-definite") - n = B.nrows() - V = VectorSpace(field, n) - def inner_product(x,y): return (B*x).inner_product(y) def jordan_product(x,y): + P = x.parent() x0 = x[0] xbar = x[1:] y0 = y[0] ybar = y[1:] z0 = inner_product(x,y) zbar = y0*xbar + x0*ybar - return V([z0] + zbar.list()) + return P((z0,) + tuple(zbar)) - super(BilinearFormEJA, self).__init__(field, - V.basis(), + # We know this is a valid EJA, but will double-check + # if the user passes check_axioms=True. + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + n = B.nrows() + standard_basis = FreeModule(ZZ, n).basis() + super(BilinearFormEJA, self).__init__(standard_basis, jordan_product, inner_product, **kwargs) @@ -2419,28 +2461,28 @@ class BilinearFormEJA(ConcreteEuclideanJordanAlgebra): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this algebra. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) if n.is_zero(): - B = matrix.identity(field, n) - return cls(B, field, **kwargs) + B = matrix.identity(ZZ, n) + return cls(B, **kwargs) - B11 = matrix.identity(field,1) - M = matrix.random(field, n-1) - I = matrix.identity(field, n-1) - alpha = field.zero() + B11 = matrix.identity(ZZ, 1) + M = matrix.random(ZZ, n-1) + I = matrix.identity(ZZ, n-1) + alpha = ZZ.zero() while alpha.is_zero(): - alpha = field.random_element().abs() + alpha = ZZ.random_element().abs() B22 = M.transpose()*M + alpha*I from sage.matrix.special import block_matrix B = block_matrix(2,2, [ [B11, ZZ(0) ], [ZZ(0), B22 ] ]) - return cls(B, field, **kwargs) + return cls(B, **kwargs) class JordanSpinEJA(BilinearFormEJA): @@ -2493,11 +2535,18 @@ class JordanSpinEJA(BilinearFormEJA): True """ - def __init__(self, n, field=AA, **kwargs): - # This is a special case of the BilinearFormEJA with the identity - # matrix as its bilinear form. - B = matrix.identity(field, n) - super(JordanSpinEJA, self).__init__(B, field, **kwargs) + def __init__(self, n, **kwargs): + # This is a special case of the BilinearFormEJA with the + # identity matrix as its bilinear form. + B = matrix.identity(ZZ, n) + + # Don't orthonormalize because our basis is already + # orthonormal with respect to our inner-product. + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + + # But also don't pass check_field=False here, because the user + # can pass in a field! + super(JordanSpinEJA, self).__init__(B, **kwargs) @staticmethod def _max_random_instance_size(): @@ -2507,14 +2556,14 @@ class JordanSpinEJA(BilinearFormEJA): return 5 @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): """ Return a random instance of this type of algebra. Needed here to override the implementation for ``BilinearFormEJA``. """ n = ZZ.random_element(cls._max_random_instance_size() + 1) - return cls(n, field, **kwargs) + return cls(n, **kwargs) class TrivialEJA(ConcreteEuclideanJordanAlgebra): @@ -2546,12 +2595,16 @@ class TrivialEJA(ConcreteEuclideanJordanAlgebra): 0 """ - def __init__(self, field=AA, **kwargs): + def __init__(self, **kwargs): jordan_product = lambda x,y: x - inner_product = lambda x,y: field(0) + inner_product = lambda x,y: 0 basis = () - super(TrivialEJA, self).__init__(field, - basis, + + # New defaults for keyword arguments + if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False + if "check_axioms" not in kwargs: kwargs["check_axioms"] = False + + super(TrivialEJA, self).__init__(basis, jordan_product, inner_product, **kwargs) @@ -2561,10 +2614,10 @@ class TrivialEJA(ConcreteEuclideanJordanAlgebra): self.one.set_cache( self.zero() ) @classmethod - def random_instance(cls, field=AA, **kwargs): + def random_instance(cls, **kwargs): # We don't take a "size" argument so the superclass method is # inappropriate for us. - return cls(field, **kwargs) + return cls(**kwargs) class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): r""" @@ -2597,8 +2650,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): have the same base ring; an error is raised otherwise:: sage: set_random_seed() - sage: J1 = random_eja(AA) - sage: J2 = random_eja(QQ,orthonormalize=False) + sage: J1 = random_eja(field=AA) + sage: J2 = random_eja(field=QQ,orthonormalize=False) sage: J = DirectSumEJA(J1,J2) Traceback (most recent call last): ... @@ -2651,8 +2704,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): EXAMPLES:: - sage: J1 = HadamardEJA(2,QQ) - sage: J2 = JordanSpinEJA(3,QQ) + sage: J1 = HadamardEJA(2, field=QQ) + sage: J2 = JordanSpinEJA(3, field=QQ) sage: J = DirectSumEJA(J1,J2) sage: J.factors() (Euclidean Jordan algebra of dimension 2 over Rational Field, @@ -2781,8 +2834,8 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): EXAMPLE:: - sage: J1 = HadamardEJA(3,QQ) - sage: J2 = QuaternionHermitianEJA(2,QQ,orthonormalize=False) + sage: J1 = HadamardEJA(3,field=QQ) + sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False) sage: J = DirectSumEJA(J1,J2) sage: x1 = J1.one() sage: x2 = x1